A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations

Engineering Analysis with Boundary Elements 82 (2017) 162–171

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations Qingsong Hua a, Yan Gu b,∗, Wenzhen Qu c, Wen Chen d, Chuanzeng Zhang e a

School of Electromechanic Engineering, Qingdao University, Qingdao 266071, PR China School of Mathematics and Statistics, Qingdao University, Qingdao 266071, PR China c School of Science, Shandong University of Technology, Zibo 255049, PR China d College of Mechanics and Materials, Hohai University, Nanjing 210098, PR China e Department of Civil Engineering, University of Siegen, Paul-Bonatz-Str. 9-11, D-57076 Siegen, Germany b

a r t i c l e

i n f o

Keywords: Generalized finite difference method Meshless method Cauchy problem Three-dimensional Helmholtz equation Inverse problem

a b s t r a c t The generalized finite difference method (GFDM) is a relatively new domain-type meshless method for the numerical solution of certain boundary value problems. The method involves a coupling between the Taylor series expansions and weighted moving least-squares method. The main idea here is to fully inherit the high-accuracy advantage of the former and the stability and meshless attributes of the latter. This paper makes the first attempt to apply the method for the numerical solution of inverse Cauchy problems associated with three-dimensional (3D) Helmholtz-type equations. Numerical results for three benchmark examples involving Helmholtz and modified Helmholtz equations in both smooth and piecewise smooth 3D geometries have been analyzed. The convergence, accuracy and stability of the method with respect to increasing the number of scatted nodes inside the whole domain and decreasing the amount of noise added into the input data, respectively, have been well-studied. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction The generalized finite difference method (GFDM) is a relatively new domain-type meshless method [1–3] for the numerical solution of boundary value problems governed by certain partial differential equations. The method circumvents or greatly eliminates the task of mesh generation which could be arduous, time-consuming and computationally expensive for certain classes of problems [4–9]. In the GFDM, the derivatives of unknown variables at a node can be approximated by a linear combination of function values with respect to its neighboring points [10]. The basis of the method was published in the early eighties by Lizska and Orkisz [11,12] and was later improved by many other authors. Now the most advanced version of the method was given by Benito et al. [10] which includes the node generation, local approximation and automatic selection of stars. In 2003, Gavete et al. [13] analyzed the influences of several factors on the accuracy of the GFDM, which can be viewed as a good guidance for using the method. Then in 2004, Benito et al. [14] proposed an h-adaptive algorithm to further improve the accuracy and efficiency of the method. Urena et al. [15] extended the method to solve third- and fourth-order partial differential equations. In a more recent study, the GFDM has been successfully ap-

∗

Corresponding authors. E-mail address: [email protected] (Y. Gu).

http://dx.doi.org/10.1016/j.enganabound.2017.06.005 Received 3 April 2017; Received in revised form 11 June 2017; Accepted 11 June 2017 0955-7997/© 2017 Elsevier Ltd. All rights reserved.

plied to problems governed by second order non-linear elliptic partial differential equations [16]. Prior to this study, the GFDM has been successfully tried for 2D and 3D problems associated with certain partial differential equations [17–24]. These problems are known as well-posed direct problems in which the Dirichlet or Neumann data on the whole boundary are known. In contrast, in inverse problems [25], one or more of the data describing the direct problem is missing. To fully determine the process, additional data must be supplied, either other boundary conditions on the same accessible part of boundary or measurements at some internal points in the domain. The inverse problems are difficult to solve due to the fact that they are ill-posed in the sense that small errors in measured data may lead to arbitrarily large changes in the numerical solution [26–28]. In this study, we investigate a numerical scheme based on the GFDM for solving the inverse Cauchy problems associated with 3D Helmholtz problems. Helmholtz-type equations arise naturally in many branches of science and engineering related to wave propagation and vibration phenomena [29,30]. It is often used to describe the vibration of a structure [31], the acoustic cavity problem [32], the radiation wave [33], and the scattering of a wave [34]. An advanced boundary element method (BEM) using the singular value decomposition (SVD) for the

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Engineering Analysis with Boundary Elements 82 (2017) 162–171

reconstruction of sound fields has been developed by Bai [25]. The vibrational velocity, sound pressure and acoustic power on the vibrating boundary comprising an enclosed space have been reconstructed by Jeon et al. [35]. Marin et al. [27] have solved the Cauchy problem for 3D Helmholtz-type equations by employing the method of fundamental solutions (MFS) in conjunction with Tikhonov regularization method. Chen and Fu [36] applied the meshless boundary particle method (BPM) for inverse Cauchy problems of inhomogeneous Helmholtz equations. It should be noted that the GFDM has recently been applied to inverse problems with great success, such as inverse biharmonic boundary value problems [17] and Cauchy problems for various partial differential equations [18]. To our knowledge, the application of the GFDM to the Cauchy problem associated with 3D Helmholtz-type equations has not been investigated yet. A brief outline of the rest of this paper is as follows. Section 2 introduces the mathematical formulation for inverse Helmholtz-type problems. The GFDM formulation and its numerical implementation for general 3D partial differential equations are presented in Section 3. Next, numerical analyzes are presented in Section 4 on three benchmark test problems with both smooth and piecewise smooth geometries. Finally, some conclusions and remarks are provided in Section 5.

Fig. 1. An irregular cloud of points and the selection of stars via distance criterion.

𝑎1

2. Mathematical formulation for inverse Helmholtz-type problems

𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕2 𝑢 𝜕2 𝑢 𝜕2 𝑢 𝜕2 𝑢 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕 𝑥𝜕 𝑦 𝜕 𝑥2 𝜕 𝑦2 𝜕 𝑧2 + 𝑎8

Consider a 3D open bounded domain Ω and assume that Ω is bounded by a surface Γ = 𝜕Ω which may consist of several segments, each being sufficiently smooth in the sense of Liapunov. We also assume that the boundary consists of two parts, Γ = Γ1 ∪ Γ2 , where Γ1 , Γ2 ≠ ∅ and Γ1 ∩ Γ2 = ∅. In this study, we refer to acoustics for the sake of the physical explanation. Hence the function u(x), which denotes the acoustical field in Ω, satisfies the following Helmholtz-type equation, namely ( 2 ) (1) ∇ ± 𝑘2 𝑢(𝒙) = 𝑓 (𝒙), 𝒙 ∈ Ω,

𝑓 𝑜𝑟 𝒙 ∈ Γ1 ,

𝑣(𝒙) = ∇𝑢(𝒙) ⋅ 𝒏(𝒙) = 𝑣̄ (𝒙)

(2) for 𝒙 ∈ Γ1 ,

(4)

where u(x, y, z) is the unknown function and f(x, y, z) is the nonhomogeneous term. In the GFDM, by utilizing the Taylor series expansions and weighted least-squares method [10,14], the derivatives of unknown variables can be approximated by a linear combination of function values with respect to some neighboring nodes which are located inside a star. First of all, an irregular cloud of points is generated in the computation domain and along the boundary. For a given ith node, which is denoted as a central node, the m nearest nodes surrounding the ith node will be found. The concept of the star then refers to a group of established nodes in relation to the central node, as shown in Fig. 1. In general each node scatted in the domain as well as along the boundary is assigned an associated star. If ui is the value of the function at the central node of the star and uj (j = 1, 2, ..., m) is the function value at the rest of node, then the Taylor series expansion around the central node can be expressed in the following form

subject to the following boundary conditions 𝑢(𝒙) = 𝑢̄ (𝒙)

𝜕2 𝑢 𝜕2 𝑢 + 𝑎9 = 𝑓 (𝑥, 𝑦, 𝑧), 𝜕 𝑥𝜕 𝑧 𝜕 𝑦𝜕 𝑧

(3)

∇2

where is the Laplace operator, k is the frequency of the acoustical field, v(x) stands for the normal velocity of the sound at a point x ∈ Γ1 , n(x) denotes the outward unit normal vector, and the barred quantities 𝑢̄ (𝒙) and 𝑣̄ (𝒙) indicate the given values specified on the boundary. In Eq. (1), the problem is known as the Helmholtz problem for the case that (∇2 + k2 )u(x) = f(x) and the modified Helmholtz problem for (∇2 − k2 )u(x) = f(x). In this study, both the cases will be considered. In the direct problems, the knowledge of the acoustical field u(x) and/or normal velocity of the sound v(x) on the boundary Γ are available and enables us to determine the acoustical field at any point inside the computational domain. In contrast, in inverse Cauchy problems, the boundary Γ1 is over-specified (see Eqs. (2) and (3)) by prescribing both the acoustical field u(x) and the normal velocity of the sound v(x) on it, whilst the remaining boundary Γ2 is under-specified since both u(x) and v(x) are unknown and have to be determined. The inverse Cauchy problem is much difficult to solve both analytically and numerically than the direct problem, since the solution does not satisfy the general conditions of well-posedness.

𝑢𝑗 = 𝑢𝑖 + ℎ𝑖𝑗

𝜕 𝑢𝑖 𝜕𝑢 𝜕𝑢 𝜕 2 𝑢𝑖 1 2 𝜕 2 𝑢𝑖 1 + 𝑘𝑖𝑗 𝑖 + 𝑙𝑖𝑗 𝑖 + ℎ2𝑖𝑗 + 𝑘𝑖𝑗 𝜕𝑥 𝜕𝑦 𝜕𝑧 2 2 𝜕 𝑥2 𝜕 𝑦2

2 𝜕 2 𝑢𝑖 𝜕 2 𝑢𝑖 𝜕 2 𝑢𝑖 1 2 𝜕 𝑢𝑖 + 𝑙𝑖𝑗 + ℎ𝑖𝑗 𝑘𝑖𝑗 + ℎ𝑖𝑗 𝑙𝑖𝑗 + 𝑘𝑖𝑗 𝑙𝑖𝑗 + ⋯⋯ 2 2 𝜕𝑧 𝜕 𝑥𝜕 𝑦 𝜕 𝑥𝜕 𝑧 𝜕 𝑦𝜕 𝑧

(5)

where the approximation is truncated after the second-order derivative, (xi ,yi ,zi ) are the coordinates of the central node, (xj ,yj ,zj ) are the coordinates of the rest of the node in the star, and hij = xj − xi , kij = yj − yi , lij = zj − zi . Writing Eq. (5) for each of the nodes in the star, it is then possible to define the function B(u) in the following manner [10,15] 2 ⎡⎛𝑢 − 𝑢 + ℎ 𝜕 𝑢𝑖 + 𝑘 𝜕 𝑢𝑖 + 𝑙 𝜕 𝑢𝑖 ⎞ ⎤ 𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 ⎢⎜ 𝑖 ⎟ ⎥ 𝜕𝑥 𝜕𝑦 𝜕𝑧 ⎟ ⎥ 𝑚 ⎢⎜ ∑ )⎥ ⎢⎜ 1 2 𝜕 2 𝑢𝑖 1 2 𝜕 2 𝑢𝑖 1 2 𝜕 2 𝑢𝑖 ⎟ ( 𝐵(𝑢) = ⎢⎜+ ℎ𝑖𝑗 2 + 𝑘𝑖𝑗 2 + 𝑙𝑖𝑗 2 ⎟𝜔 ℎ𝑖𝑗 , 𝑘𝑖𝑗 , 𝑙𝑖𝑗 ⎥ , (6) 2 2 2 𝜕𝑧 𝜕𝑥 𝜕𝑦 ⎟ ⎥ 𝑗=1 ⎢⎜ 2 2 2 ⎢⎜ ⎥ 𝜕 𝑢𝑖 𝜕 𝑢𝑖 𝜕 𝑢𝑖 ⎟ ⎢⎜+ ℎ𝑖𝑗 𝑘𝑖𝑗 ⎟ ⎥ + ℎ𝑖𝑗 𝑙𝑖𝑗 + 𝑘𝑖𝑗 𝑙𝑖𝑗 ⎣⎝ ⎦ 𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧 𝜕𝑦𝜕𝑧 ⎠

where 𝜔(hij ,kij ,lij ) is the denominated weighting function at point (xj ,yj ,zj ). In all the examples considered in this paper, the weighting function is chosen as ( )2 ( 𝑑 )3 ( 𝑑 )4 ⎧ 𝑖𝑗 𝑖𝑗 ( ) ⎪1 − 6 𝑑𝑖𝑗 + 8 𝑑𝑚 − 3 𝑑𝑚 , 𝑑𝑖𝑗 ≤ 𝑑𝑚, 𝑑𝑚 𝜔 ℎ𝑖𝑗 , 𝑘𝑖𝑗 , 𝑙𝑖𝑗 = ⎨ (7) ⎪0, 𝑑𝑖𝑗 > 𝑑𝑚. ⎩

3. The generalized finite difference method (GFDM) and its numerical implementation 3.1. GFDM for second-order partial differential equations Without loss of generality, let us consider a general 3D problem governed by the following second-order partial differential equation 163

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where dij denotes the distance between nodes (xi ,yi ,zi ) and (xj ,yj ,zj ), dm is the distance between (xi ,yi ,zi ) and the farthest node in the star. Of course other choices are possible, as has been illustrated in Refs. [17,37]. The weighting function is used here to indicate the importance of the approximations at different nodes, which means that the Taylor series approximation is more important if the node is closer to the center of the star. To minimize the residual function B(u) with respect to the unknown derivatives at the central point (xi ,yi ,zi ) yields the following linear equation system 𝑨𝑫 𝑢 = 𝒃,

Theoretically speaking [18], for 3D partial differential equations, the minimum number of nodes included into a star is 9 to ensure the solvability of the Eq. (8). To avoid ill-conditioning, the distribution of these nodes should ideally be arranged in a sort of regular polygonal or star formation about the central node. However, this is difficult to achieve in practice. To overcome this problem, it is usual to include more nodes in the star and to seek instead an approximation to the functional derivative by the moving least-squares method. A guidance for the choice of these supporting nodes can be found in Refs. [3,38]. It is noted that the vector b in Eq. (11) can be reformulated as the following form

(8)

𝒃 = 𝑩𝑼 ,

where { }𝑇 𝜕 𝑢𝑖 𝜕 𝑢𝑖 𝜕 𝑢𝑖 𝜕 2 𝑢𝑖 𝜕 2 𝑢𝑖 𝜕 2 𝑢𝑖 𝜕 2 𝑢𝑖 𝜕 2 𝑢𝑖 𝜕 2 𝑢𝑖 𝑫𝑢 = , , , , , , , , , 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕 𝑥2 𝜕 𝑦2 𝜕 𝑧2 𝜕 𝑥𝜕 𝑦 𝜕 𝑥𝜕 𝑧 𝜕 𝑦𝜕 𝑧

(12)

where U =[ui , u1 , u2 , …, um is the function value at the nodes inside the star. According to Eqs. (8) and (12), the vector Du in Eq. (9) can be expressed by the following equation ]T

(9)

is the partial derivative vector at the point (xi ,yi ,zi ), and 𝑚 ⎡∑ 2 2 ⎢ ℎ𝑖𝑗 𝑤 ⎢ 𝑗=1 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 𝑨=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝑚 ∑ 𝑗=1 𝑚 ∑ 𝑗=1

ℎ𝑖𝑗 𝑘𝑖𝑗 𝑤2 𝑘2𝑖𝑗 𝑤2

𝑚 ∑ 𝑗=1 𝑚 ∑

ℎ𝑖𝑗 𝑙𝑖𝑗 𝑤2 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤2

𝑗=1 𝑚 ∑ 𝑗=1

2 2 𝑙𝑖𝑗 𝑤

𝑚 ℎ3 ∑ 𝑖𝑗

2

𝑤2

𝑗=1 𝑚 ℎ2 𝑘 ∑ 𝑖𝑗 𝑖𝑗 𝑗=1 𝑚 ∑

2 ℎ2𝑖𝑗 𝑙𝑖𝑗

𝑗=1 𝑚 ∑ 𝑗=1

2 ℎ4𝑖𝑗 4

𝑤2 𝑤2

𝑤2

𝑚 ℎ 𝑘2 ∑ 𝑖𝑗 𝑖𝑗 𝑗=1 𝑚 ∑

2 𝑘3𝑖𝑗 2

𝑤2

𝑗=1 𝑚 𝑘2 𝑙 ∑ 𝑖𝑗 𝑖𝑗

2

𝑗=1 𝑚 ℎ2 𝑘2 ∑ 𝑖𝑗 𝑖𝑗 𝑗=1 𝑚 ∑ 𝑗=1

4 𝑘4𝑖𝑗 4

𝑤2

𝑤2 𝑤2

𝑤2

𝑚 ℎ 𝑙2 ∑ 𝑖𝑗 𝑖𝑗 𝑗=1 𝑚 ∑

2 2 𝑘𝑖𝑗 𝑙𝑖𝑗

2

𝑗=1 𝑚 𝑙3 ∑ 𝑖𝑗

2 4

2 𝑘2𝑖𝑗 𝑙𝑖𝑗

4

𝑗=1 𝑚 𝑙4 ∑ 𝑖𝑗 𝑗=1

4

𝑤2

𝑤2

𝑗=1 𝑚 ℎ2 𝑙 2 ∑ 𝑖𝑗 𝑖𝑗 𝑗=1 𝑚 ∑

𝑤2

𝑤2 𝑤2

𝑤2

𝑆𝑌 𝑀

𝑚 ∑ 𝑗=1 𝑚 ∑ 𝑗=1 𝑚 ∑

ℎ2𝑖𝑗 𝑘𝑖𝑗 𝑤2 ℎ𝑖𝑗 𝑘2𝑖𝑗 𝑤2

ℎ𝑖𝑗 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤2

𝑗=1 𝑚 ∑ 𝑗=1 𝑚 ∑

ℎ3𝑖𝑗 𝑘𝑖𝑗 2 𝑘3𝑖𝑗 ℎ𝑖𝑗

𝑗=1 𝑚 ℎ ∑ 𝑗=1 𝑚 ∑ 𝑗=1

2

𝑤2 𝑤2

2 𝑖𝑗 𝑘𝑖𝑗 𝑙𝑖𝑗

2

𝑤2

ℎ2𝑖𝑗 𝑘2𝑖𝑗 𝑤2

𝑚 ∑ 𝑗=1 𝑚 ∑

ℎ2𝑖𝑗 𝑙𝑖𝑗 𝑤2

ℎ𝑖𝑗 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤2

𝑗=1 𝑚 ∑

2 2 ℎ𝑖𝑗 𝑙𝑖𝑗 𝑤 𝑗=1 𝑚 ℎ3 𝑙 ∑ 𝑖𝑗 𝑖𝑗 2

𝑗=1 𝑚 ℎ ∑

2

2 𝑖𝑗 𝑘𝑖𝑗 𝑙𝑖𝑗

2

𝑗=1 𝑚 𝑙3 ℎ ∑ 𝑖𝑗 𝑖𝑗 𝑗=1 𝑚 ∑

𝑤

2

𝑤2

𝑤2

ℎ2𝑖𝑗 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤2

𝑗=1 𝑚 ∑ 𝑗=1

2 2 ℎ2𝑖𝑗 𝑙𝑖𝑗 𝑤

𝑚 ∑

⎤ ℎ𝑖𝑗 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤2 ⎥ ⎥ ⎥ ⎥ 𝑘2𝑖𝑗 𝑙𝑖𝑗 𝑤2 ⎥ 𝑗=1 ⎥ 𝑚 ∑ ⎥ 2 2 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤 ⎥ ⎥ 𝑗=1 ⎥ 𝑚 ℎ2 𝑘 𝑙 ∑ 𝑖𝑗 𝑖𝑗 𝑖𝑗 2 ⎥ 𝑤 ⎥ 2 𝑗=1 ⎥ 3 𝑚 ⎥ ∑ 𝑘𝑖𝑗 𝑙𝑖𝑗 2 ⎥ 𝑤 2 ⎥ 𝑗=1 ⎥ 𝑚 𝑙3 𝑘 ∑ 𝑖𝑗 𝑖𝑗 2 ⎥ 𝑤 ⎥ 2 𝑗=1 ⎥ 𝑚 ⎥ ∑ 2 2 ⎥ ℎ𝑖𝑗 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤 ⎥ 𝑗=1 ⎥ 𝑚 ∑ 2 2 ⎥ ℎ𝑖𝑗 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤 ⎥ 𝑗=1 ⎥ 𝑚 ∑ ⎥ 2 2 𝑘2𝑖𝑗 𝑙𝑖𝑗 𝑤 ⎥ ⎦ 𝑗=1 𝑗=1 𝑚 ∑

(10) ⎡ 𝜕 𝑢𝑖 ⎤ ⎢ 𝜕𝑥 ⎥ ⎢ 𝜕𝑢 ⎥ ⎡ 𝑢𝑖 ⎤ 𝑖 ⎥ ⎢ ⎢𝑢 ⎥ ⎢ 𝜕𝑦 ⎥ ⎢ 1⎥ −1 −1 ⎢ ⎥ 𝑫 𝑢 = 𝜕 𝑢𝑖 = 𝑨 𝒃 = 𝑨 𝑩 𝑼 = 𝑬 ⎢ 𝑢2 ⎥, ⎢ ⎥ ⎢ ⋯⎥ ⎢ 𝜕𝑧 ⎥ ⎢ ⎥ ⎢ ⋯ ⎥ ⎣ 𝑢𝑚 ⎦ ⎢ 𝜕2 𝑢 ⎥ 𝑖 ⎢ ⎥ ⎣ 𝜕 𝑦𝜕 𝑧 ⎦

is a symmetrical matrix of 9 × 9 which contains geometric information about distribution of the selected nodes around the central node, and the vector b is 9 × 1 which can be expressed as 𝑚 ⎡∑ ( 𝑢𝑗 ⎢ ⎢ 𝑗=1 𝑚 ⎢∑ ( ⎢ 𝑢 ⎢ 𝑗=1 𝑗 ⎢∑ 𝑚 ⎢ (𝑢 𝑗 ⎢ ⎢ 𝑗=1 𝑚 ⎢∑ ( 𝑢𝑗 ⎢ ⎢ 𝑗=1 𝑚 ⎢∑ ( 𝑢 𝒃=⎢ ⎢ 𝑗=1 𝑗 ⎢∑ 𝑚 ⎢ (𝑢 𝑗 ⎢ ⎢ 𝑗=1 𝑚 ∑ ⎢ ( 𝑢𝑗 ⎢ ⎢ 𝑗=1 𝑚 ⎢∑ ( ⎢ 𝑢 ⎢ 𝑗=1 𝑗 ⎢∑ 𝑚 ⎢ (𝑢 𝑗 ⎢ ⎣ 𝑗=1

⎤ ⎥ ⎥ ⎥ ) − 𝑢𝑖 𝑘𝑖𝑗 𝑤2 ⎥ ⎥ ⎥ ) ⎥ 2 − 𝑢𝑖 𝑙𝑖𝑗 𝑤 ⎥ ⎥ ) 2 2 ⎥ − 𝑢𝑖 ℎ𝑖𝑗 𝑤 ∕2 ⎥ ⎥ ) 2 2 ⎥ − 𝑢𝑖 𝑘𝑖𝑗 𝑤 ∕2 ⎥. ⎥ ⎥ )2 2 ⎥ − 𝑢𝑖 𝑙𝑖𝑗 𝑤 ∕2 ⎥ ⎥ ) ⎥ − 𝑢𝑖 ℎ𝑖𝑗 𝑘𝑖𝑗 𝑤2 ⎥ ⎥ ⎥ ) 2 ⎥ − 𝑢𝑖 ℎ𝑖𝑗 𝑙𝑖𝑗 𝑤 ⎥ ⎥ ) − 𝑢𝑖 𝑘𝑖𝑗 𝑙𝑖𝑗 𝑤2 ⎥⎥ ⎦ ) − 𝑢𝑖 ℎ𝑖𝑗 𝑤2

(13)

where E =A − 1 B, which can be calculated using the moving least-squares method. According to the above analysis, the explicit expressions of partial derivatives Du at the central node have been obtained, with respect to the function values at the nodes inside the star. Substituting Eq. (13) into the partial differential Eq. (4), then the star equation corresponding to the central node (xi ,yi ,zi ) is obtained as 𝑚 ∑ 𝑚𝑖 𝑢𝑖 + 𝑚𝑗 𝑢𝑗 = 𝑓 (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ), (14)

(11)

𝑗=1 𝑗≠𝑖

where the coefficients mi and mj can be immediately obtained by using Eqs. (4) and (13). This procedure should be repeated at each point inside the computational domain. By enforcing the satisfaction of the boundary conditions at the boundary nodes and the governing equation at the interior nodes, a system of linear algebraic equations, with 164

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respect to all unknown values {𝑢𝑖 }𝑁 , can now be formed, where N is 𝑖=1 the total number of nodes in the computational domain. On solving this system of equations, the approximated values of {𝑢𝑖 }𝑁 are obtained 𝑖=1 and the corresponding partial derivatives may easily be calculated using Eq. (13). It is interesting to note that the concept of star in the GFDM yields a sparse matrix system in contrast to a fully populated matrix in, for example, the MFS and BEM. Therefore, the GFDM is very efficient to analyze problems defined in high dimensions and complex geometries. 3.2. GFDM for inverse Cauchy Helmholtz-type equations For 3D inverse Cauchy problems, assume that the boundary of the computational domain consists of two parts, Γ = Γ1 ∪ Γ2 , where Γ1 ,Γ2 ≠ ∅ and Γ1 ∩ Γ2 = ∅. The boundary Γ1 is over-specified by prescribing both the acoustics field and the sound velocity, whilst the remaining boundary Γ2 is under-specified since both the acoustics field and the sound velocity are unknown. To solve the problem numerically, nb = n1 + n2 boundary nodes and ni interior nodes are distributed along the boundary and inside the computational domain, respectively. Here, n1 and n2 denote the number of boundary nodes distributed along boundaries Γ1 and Γ2 , respectively. To enforce the satisfaction of the governing equation at ni interior nodes and the boundary conditions at n1 boundary nodes yields the following ni + 2 × n1 linear algebraic equations: ⎡⎧ 𝑢̄ (𝒙1 ) ⎫ ⎤ ⎢⎪ ⎪ ⎥ ⎢⎨ ⋮ ⎬ ⎥ ⎢⎪𝑢̄ (𝒙𝑛1 )⎪ ⎥ ⎭ ⎥ ⎢⎩ ⎡𝑢 1 ⎤ [ ] ⎢⎧ ⎥ ⎛ 𝐴1 𝑛 ×𝑁 ⎞⎢𝑢 ⎥ 1) ⎫ 𝑣 ̄ ( 𝒙 ⎢⎪ ⎥ 1 [ ] ⎜ ⎟⎢ 2 ⎥ ⎢⎨ ⋮ ⎪ ⎥ = ⎜ 𝐴2 𝑛 ×𝑁 ⎟⎢𝑢3 ⎥ , 1 [ ] ⎢⎪ 𝑛 ⎬ ⎥ ⎜𝐴 ⎟⎢⋮ ⎥ ⎢⎩𝑣̄ (𝒙 1 )⎪ ⎥ 3 ⎭ ⎝ 𝑛𝑖 ×𝑁 ⎠⎢ ⎥ ⎢ ⎥ ⎣𝑢𝑁 ⎦𝑁×1 ⎢⎧ 𝑓 (𝒙𝑛1 +1 ) ⎫⎥ ⎪⎥ ⎢⎪ ⋮ ⎬⎥ ⎢⎨ ⎢⎪𝑓 (𝒙𝑛1 +𝑛𝑖 )⎪⎥ ⎭⎦(𝑛 +2×𝑛 )×1 ⎣⎩ 𝑖 1

Fig. 2. Geometry of the problem and the configuration of the nodes distribution.

where b is the exact boundary data, rand is a random number and its range is − 1 ≤ rand ≤ 1, and 𝛿 stands for the level of noise. Unless otherwise specified, the measure of the accessible (or measured) boundary is defined as Γ1 = BL × Γ in which

(15)

𝐵𝐿 =

1 ×𝑁

⎡1 ⎢ 0 =⎢ ⎢⋮ ⎢0 ⎣

0 1 ⋮ 0

0 0 ⋮ …

⋯ ⋯ ⋱ 1

0⎤ ⎥ 0⎥ ⋮⎥ …⎥⎦

,

𝐵 𝐿 ∈ (0, 1],

(18)

stands for the ratio parameter of the accessible boundary. In order to evaluate the performance of the numerical method, an L2 error norm is defined as √ √ √𝑁 /√ 𝑁 √∑ √∑ 2 √ √ 𝐼 𝑖 2, 𝑖 𝑖 Relative Error = (𝐼numerical − 𝐼exact ) (19) exact

where xi = (xi ,yi ,zi ) denotes the coordinates of the ith nodes, [ ] 𝐴1 𝑛

measure(Γ1 ) , measure(Γ)

(16)

𝑖=1

𝑖=1

𝑖 𝑖 where 𝐼numerical and 𝐼exact denote numerical and analytical solutions at the ith test point, respectively, N is the total number of collocation points at which both the numerical and exact solutions are evaluated. For the ease of comparison and validation of numerical results, the analytical solutions are taken to be

𝑛1 ×𝑁

and A2 , A3 are matrices of n1 × N and ni × N, respectively, which can be obtained straightforward by using Eq. (14). Combining the above ni + 2 × n1 linear algebraic equations will form the final over-determined matrix system. Finally, we can obtain the approximate nodal values u(x, y, z) at each point by solving this matrix system. For an inverse problem, the interpolation matrix is, in general, severely ill-conditioned. In the past, it is customary to use various regularization techniques [39–42], such as the Tikhonov regularization and truncated singular value decomposition (TSVD) methods, for solving stably and accurately ill-conditioned matrix equations. In our computations we use the moving least-squares method, which can be regarded as one kind of regularization method, to relieve the problems of ill-conditioned matrices and to stabilize the numerical scheme.

𝑢(𝑥, 𝑦, 𝑧) = 𝑐𝑜𝑠(𝑥 + 2𝑦 + 2𝑧),

(20)

for Helmholtz equation, and 𝑢(𝑥, 𝑦, 𝑧) = exp(𝑥 + 𝑦 + 𝑘𝑧),

(21)

for modified Helmholtz equation. 4.1. Test problem 1: Helmholtz problems 𝛾 ≡ Δ + k2 in a cubic domain First, we consider the Helmholtz problem in a cubic domain 3 3 Ω √= (0, 1) ⊂R . The analytical solution is shown in Eq. (20) with 𝑘 = 9, which is an example used in Ref. [27]. The problem sketch and the configuration of the nodes distribution of the GFDM are depicted in Fig. 2. Here the bottom surface {(x, y, z)|z = 0} and the right-hand side surface {(x, y, z)|y = 1} are under-specified where both the acoustical field and the normal velocity of the sound are unknown and have to be determined. To solve the problem numerically, N = 896 evenly distributed points are chosen inside the whole domain. Fig. 3(a–d) illustrates the analytical (solid lines) and the numerical results (dashed lines) for the acoustical field obtained on the bottom surface, using various levels of noise added into the input data. It

4. Results and discussion In this section, three benchmark numerical examples associated with 3D Helmholtz and modified Helmholtz equations, with both smooth and piecewise smooth geometries, are presented to verify the methods developed above. The effect of the method, as well as the stability of the scheme with respect to the noise added into the input data, is carefully investigated. In our test cases, the simulated noisy data are generated using the following formula ( ) 𝛿 𝑏̃ = 𝑏 1 + 𝑟𝑎𝑛𝑑 × (17) , 100 165

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Fig. 3. Distributions of the calculated acoustical fields on the bottom surface, with (a) 0% noisy, (b) 1% noisy, (c) 2% noisy, and (d) 3% noisy added into the input data.

can be seen that the numerical results retrieved for sound are in good agreement with their corresponding analytical solutions, even with a relatively large amount of noise (3%). We can also observed that, as expected, the numerical solutions converge to their corresponding analytical solutions as the amount of noise decreases, indicating that the GFDM yields accurate and stable numerical results for noisy data. It is worth mentioning that, although not presented herein, similar conclusions have been drawn for the other examples investigated in this study. Fig. 4(a−d) illustrates the relative error curves of the computed acoustical field on the right-hand side surface obtained using various levels of noise added into the input data. As shown in Fig. 4(a), the proposed numerical scheme is extremely accurate for problems with exact input data, with the maximum error being 15 × 10 − 3 . As shown in Fig. 4(b−d), even for a relatively high amount of noise added into the input data, the numerical results retrieved for the acoustical field represent good approximations for their analytical values. Hence the GFDM, in conjunction with the moving least-squares method, provides stable numerical solutions to the inverse Cauchy problems governed by Helmholtz-type equations. Next, we investigate the influence of the number m of supporting nodes inside a star on the accuracy and stability of the numerical solutions. To do so, the relative error curves of the computed acoustical field on the bottom surface with 1% noise added into the data, as a function of the number m of supporting nodes, is illustrated in Fig. 5. It can be seen that the numerical results agree quite well with the corre-

sponding analytical solutions for various numbers of supporting nodes. And, in general, the accuracy of the evaluations is relatively insensitive to the number m. To investigate the convergence of the numerical results, the error distributions for acoustical field, obtained using N = 896 and N = 2000 nodes scatted inside the whole computational domain, are also illustrated in Fig. 5. It can be seen that the numerical results converge to their corresponding analytical solutions as the number of nodes increases. Similar results have been obtained for the other examples analyzed and these indicate that the proposed algorithm yields very accurate numerical results for noisy data. Fig. 6 displays the relative errors of the computed acoustical field on the bottom surface of the structure, with 3% noise added into the input data. Numerical results obtained using the MFS proposed by Marin [27] are also given for the purpose of comparison. In the MFS, N = 600 evenly distributed source points were chosen on the fictitious boundary Ω = ( − 5, 5)3 ⊂R3 , as used in Ref. [27]. As shown in Fig. 6, the GFDM results agree pretty well with those of the MFS. 4.2. Test problem 2: modified Helmholtz problems 𝛾 ≡ Δ − k2 in a 3D peanut-shaped domain Next, we consider the modified Helmholtz problem through a 3D peanut-shaped domain represented by the following parametric surface 𝒙(𝜙, 𝜃) = (𝑓 (𝜙) sin 𝜙 cos 𝜃, 𝑓 (𝜙) sin 𝜙 sin 𝜃, 𝑓 (𝜙) cos 𝜙), 166

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0.2

(a)

0.4

0.6

0.8

(b) 0

0.8 -0.005

0.6 -0.01

0.4 -0.015

0.2

0.2

0.4

0.6

-0.02

0.8

(c)

(d)

Fig. 4. The error distribution for computed acoustical field on the right-hand side surface, with (a) 0% noisy, (b) 1% noisy, (c) 2% noisy, and (d) 3% noisy added into the input data.

5.5

√ acoustical field is taken to be 𝑘 = 6. To solve the problem numerically, N = 1600 evenly distributed points are chosen inside the whole domain. The accessible boundary is specified as Γ1 ={x(𝜙,𝜃)|0 < 𝜙 < 𝜋 × BL}, where BL changes between (0.5, 1). Fig. 8 presents the numerical results for acoustical field on the underspecified surface, with BL = 0.75 and various levels of data noise. It can be observed the numerical solutions converge to their corresponding analytical solutions as the amount of noise decreases. Even for a relatively high amount of noise (3%) added into the data, the numerical results still agree quite well with the corresponding analytical solutions, indicating that the GFDM, together with the moving least-squares method, can yield accurate and stable numerical results for noisy data. Fig. 9 illustrates the relative error curves of acoustical field on nodes inside the whole domain, as functions of various levels of noisy Cauchy data. In order to analyze the sensitivity of the proposed method with respect to the length of the accessible boundary Γ1 , the parameter BL here changes between 0.5–0.98. As can be seen in Fig. 9, it can be seen that the numerical results agree quite well with the corresponding analytical solutions, even for a relatively high amount of noise level (2%), and the numerical results converge to the analytical solutions as the amount of data noise decreases. Further, as expected, the numerical results improve significantly as the length of the accessible boundary increases, i.e. BL increases.

10-3 N=896 nodes inside the whole domain

Relative error

5

N=2000 nodes inside the whole domain

4.5

4

3.5

3

2.5 65

70

75

80

85

90

95

100

The number m of supporting nodes inside a star Fig. 5. Relative error curves obtained using 1% noise added into the input data, as functions of various numbers m of supporting nodes inside a star.

√ where 𝜙 ∈ [0, 𝜋], 𝜃 ∈ [0, 2𝜋) and 𝑓 (𝜙) =

cos(2𝜙) +

√ 1.5 − sin2 (2𝜙).

The problem sketch and the configuration of the nodes distribution of the GFDM are depicted in Fig. 7. In this example, the frequency of the 167

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(a)

(b)

Fig. 6. The error distribution for computed acoustical field on the bottom surface by using (a) the proposed GFDM and (b) the MFS proposed in Ref. [27].

0.6 0.3 0 -1

-0.3 -0.6 1.5

0 1

0.5

0

-0.5

-1

-1.5

z

1

o (a) Geometry of the problem

y

x

(b) Nodes distribution

Relative errors

Acoustical field

Fig. 7. Geometry of the problem (a) and the configuration of the nodes distribution (b).

The number of BL

Nodes on the under-specified surface

Fig. 9. Relative error curves for acoustical field on nodes inside the whole domain with respect to different length of the accessible boundary (BL).

Fig. 8. Numerical results for acoustical field on boundary points along under-specified surface, with BL = 0.75 and various levels of data noise. 168

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1 0 -1 5 3

5 1

3 1

-1

-1

-3

z

-3 -5

-5

o (a) Geometry of the problem

x

y (b) Nodes distribution

Fig. 10. Geometry of the problem (a) and the configuration of the nodes distribution (b).

(a)

(b)

(c)

(d)

Fig. 11. Relative error surfaces for example 3 with BL = 0.8, where the GFDM nodes are 800 for (a), 1500 for (b), 2500 for (c), and 5000 for (d), respectively.

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Table 1 Convergence study of the acoustical field u and its derivative 𝜕 u/𝜕 x at 300 evenly distributed inner points for example 3 with BL = 0.8. Relative error

Mesh 1 (800 nodes)

Mesh 2 (1500 nodes)

Mesh 3 (2500 nodes)

u 𝜕 u/𝜕 x

8.573E-03 3.937E-02

2.815E-03 8.162E-03

6.842E-4 3.781E-3

Acknowledgments The work described in this paper was supported by the National Natural Science Foundation of China (nos. 11402075, 11302069, 71571108), Projects of International (Regional) Cooperation and Exchanges of NSFC (no. 71411130215), the China Postdoctoral Science Foundation (nos. 2015M570572, 2015M570569, 2016T90608), and the Qingdao Postdoctoral Application Research Project (nos. 2015138, 15-9-1-49-jch).

4.3. Test problem 3: modified Helmholtz problems 𝛾 ≡ Δ − k2 in a ring torus

References

Finally, we consider the modified Helmholtz problem in a ring torus domain. The ring torus bounds a solid known as a toroid. The adjective toroidal can be applied to tori, toroids or, more generally, any ring shape in toroidal inductors and transformers. Real world examples of toroidal objects include doughnuts, inner tubes, many lifebuoys, O-rings, and vortex rings. In mathematics, a torus can be defined parametrically by 𝒙(𝜙, 𝜃) = (𝑓 (𝜙) cos 𝜃, 𝑓 (𝜙) sin 𝜃, 𝑟 sin 𝜙),

[1] Karageorghis A, Smyrlis YS. Matrix decomposition MFS algorithms for elasticity and thermo-elasticity problems in axisymmetric domains. J Comput Appl Math 2007;206(2):774–95. [2] Chen CS, Muleshkov AS, Golberg MA, Mattheij RMM. A mesh-free approach to solving the axisymmetric Poisson’s equation. Numer Methods Partial Differ Equ 2005;21(2):349–67. [3] Benito JJ, Urena F, Gavete L. Solving parabolic and hyperbolic equations by the generalized finite difference method. J Comput Appl Math 2007;209(2):208–33. [4] Cheng AHD, Cheng DT. Heritage and early history of the boundary element method. Eng Anal Bound Elem 2005;29(3):268–302. [5] Marin L, Lesnic D. The method of fundamental solutions for nonlinear functionally graded materials. Int J Solids Struct 2007;44(21):6878–90. [6] Sarler B. Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. Eng Anal Bound Elem 2009;33(12):1374–82. [7] Marin L. An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation. Comput Mech 2010;45(6):665–77. [8] Liu GR, Gu YT. A meshfree method: meshfree weak–strong (MWS) form method, for 2-D solids. Comput Mech 2003;33(1):2–14. [9] Karageorghis A, Fairweather G. The method of fundamental solutions for axisymmetric elasticity problems. Comput Mech 2000;25(6):524–32. [10] Benito JJ, Urena F, Gavete L. Influence of several factors in the generalized finite difference method. Appl Math Model 2001;25(12):1039–53. [11] Liszka T. An interpolation method for an irregular net of nodes. Int J Numer Methods Eng 1984;20(9):1599–612. [12] Liszka T, Orkisz J. The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput Struct 1980;11(1):83–95. [13] Gavete L, Gavete ML, Benito JJ. Improvements of generalized finite difference method and comparison with other meshless method. Appl Math Model 2003;27(10):831–47. [14] Benito JJ, Urena F, Gavete L, Alvarez R. An h-adaptive method in the generalized finite differences. Comput Methods Appl Mech Eng 2003;192(5–6):735–59. [15] Ureña F, Salete E, Benito JJ, Gavete L. Solving third- and fourth-order partial differential equations using GFDM: application to solve problems of plates. Int J Comput Math 2012;89(3):366–76. [16] Gavete L, Ureña F, Benito JJ, García A, Ureña M, Salete E. Solving second order non-linear elliptic partial differential equations using generalized finite difference method. J Comput Appl Math 2017;318:378–87. [17] Fan CM, Huang YK, Li PW, Chiu CL. Application of the generalized finite-difference method to inverse biharmonic boundary-value problems. Numer Heat Transf B 2014;65(2):129–54. [18] Fan CM, Li PW, Yeih WC. Generalized finite difference method for solving two-dimensional inverse Cauchy problems. Inverse Probl Sci Eng 2015;23(5):737–59. [19] Yang C, Tang DL, Atluri S. Three-dimensional carotid plaque progression simulation using mesh less generalized finite difference method based on multi-year MRI patient-tracking data. Comp Model Eng Sci 2010;57(1):51–76. [20] Gavete L, Urena F, Benito JJ, Salete E. A note on the dynamic analysis using the generalized finite difference method. J Comput Appl Math 2013;252:132–47. [21] Salete E, Benito JJ, Ureña F, Gavete L, Ureña M, García A. Stability of perfectly matched layer regions in generalized finite difference method for wave problems. J Comput Appl Math 2017;312:231–9. [22] Hosseini SM. Shock-induced two dimensional coupled non-Fickian diffusion–elasticity analysis using meshless generalized finite difference (GFD) method. Eng Anal Bound Elem 2015;61:232–40. [23] Hosseini SM. Application of a hybrid mesh-free method for shock-induced thermoelastic wave propagation analysis in a layered functionally graded thick hollow cylinder with nonlinear grading patterns. Eng Anal Bound Elem 2014;43:56–66. [24] Hosseini SM. Application of a hybrid mesh-free method based on generalized finite difference (GFD) method for natural frequency analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotubes. Comp Model Eng Sci 2013;95(1):1–29. [25] Bai MR. Application of BEM (boundary element method)‐based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries. J Acoust Soc Am 1992;92(1):533–49. [26] Gu Y, Chen W, Zhang C, He X. A meshless singular boundary method for three-dimensional inverse heat conduction problems in general anisotropic media. Int J Heat Mass Transf 2015;84(0):91–102. [27] Marin L. A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations. Appl Math Comput 2005;165(2):355–74.

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where 𝜙 and 𝜃 both vary in the interval [0, 2𝜋) and f(𝜙) = R + rcos (𝜙). R = 4 and r = 1 (also known as the “major radius” and “minor radius”) denote the distance from the center of the rube to the center of the torus and the radius of the tube, respectively. The geometry and the GFDM model for this problem are shown in Fig. 10. The accessible boundary is specified as Γ1 ={x(𝜙,𝜃)|0 ≤ 𝜃 < 2𝜋 × BL}, where BL here is taken to be 0.8. To illustrate the convergence of the proposed method, Fig. 11 displays the relative error surfaces in a square domain using 800, 1500, 2500, and 5000 nodes with 2% noise added into the input data, respectively, where the error surfaces were yielded at 30 × 30 calculation points uniformly-spacing over the square {(x, y, 0)|3.6 ≤ x ≤ 4.4, − 0.4 ≤ y ≤ 0.4}. Table 1 shows the relative errors of the acoustical field and its derivative 𝜕 u/𝜕 x at 300 calculation points evenly distributed inside the whole domain, with 2% noise added into the input data. We can see from Fig. 11 and Table 1 that the present GFDM is stable, accurate, and rapidly convergent as the number of nodes increases. It is also observed that the GFDM results with only 800 nodes are quite accurate for this 3D case. The size of resulting system of linear algebraic equations is quite small. It is worth mentioning that, although not presented herein, similar conclusions have been drawn for the other examples investigated in this study. Hence the GFDM, in conjunction with the moving least-squares method, is computationally very efficient and stable to the inverse Cauchy problems associated with Helmholtz equations.

5. Conclusions This study makes the first attempt to apply the meshless GFDM for the numerical solution of inverse Cauchy problems associated with the three-dimensional Helmholtz-type equations. Numerical results for three benchmark examples involving Helmholtz and modified Helmholtz equations in both smooth and piecewise smooth 3D geometries have been analyzed. The numerical results obtained illustrate that the proposed scheme is computationally efficient, accurate, stable with respect to decreasing the amount of noise added into the input data and convergent with respect to increasing the number of nodes. Overall, it is expected that, in comparison with existing methods for solving numerically inverse Cauchy problems associated with Helmholtz equations, the proposed scheme could be considered as a competitive alternative. The proposed numerical procedure can also be applied to inverse problems associated with other elliptic partial differential equations, such as the elasticity and electromagnetic equations, which are currently under investigation. 170

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[28] Karageorghis A, Lesnic D, Marin L. The method of fundamental solutions for solving direct and inverse Signorini problems. Comput Struct 2015;151:11–19. [29] Liu GR, Chen XL. A mesh-free method for static and free vibration analyses of thin plates of complicated shape. J Sound Vib 2001;241(5):839–55. [30] Liu GR, Wang XJ, Xi ZC. Elastodynamic responses of an immersed to a Gaussian beam pressure. J Sound Vib 2000;233(5):813–33. [31] Beskos DE. Boundary element methods in dynamic analysis. Appl Mech Rev 1987;40(1):1–23. [32] Chen JT, Chang MH, Chen KH, Lin SR. The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function. J Sound Vib 2002;257(4):667–711. [33] Harari I, Barbone E, Slavutin M, Shalom R. Boundary infinite elements for the Helmholtz equation in exterior domains. Int J Numer Methods Eng 1998;41:1105–31. [34] Hall WS, Mao XQ. A boundary element investigation of irregular frequencies in electromagnetic scattering. Eng Anal Bound Elem 1995;16(3):245–52. [35] Jeon I-Y, Ih J-G. On the holographic reconstruction of vibroacoustic fields using equivalent sources and inverse boundary element method. J Acoust Soc Am 2005;118(6):3473–82.

[36] Chen W, Fu ZJ. Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations. J Mar Sci Technol 2009;17(3):157–63. [37] Benito JJ, Urena F, Salete E, Muelas A, Gavete L, Galindo R. Wave propagation in soils problems using the generalized finite difference method. Soil Dyn Earthq Eng 2015;79:190–8. [38] Chan HF, Fan CM, Kuo CW. Generalized finite difference method for solving two-dimensional non-linear obstacle problems. Eng Anal Bound Elem 2013;37(9):1189–96. [39] Marin L. Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity. Int J Solids Struct 2010;47(24):3326–40. [40] Karageorghis A, Lesnic D. Application of the MFS to inverse obstacle scattering problems. Eng Anal Bound Elem 2011;35(4):631–8. [41] Qin HH, Wen DW. Tikhonov type regularization method for the Cauchy problem of the modified Helmholtz equation. Appl Math Comput 2008;203(2):617–28. [42] Fu Z-J, Chen W, Gu Y. Burton–Miller-type singular boundary method for acoustic radiation and scattering. J Sound Vib 2014;333(16):3776–93.

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